Question:
If the matrix A is both symmetric and skew symmetric, then
A. A is a diagonal matrix
B. A is a zero matrix
C. A is a square matrix
D. None of these
Solution:
Answer: B
If A is both symmetric and skew-symmetric matrix, then we should have
$A^{\prime}=A$ and $A^{\prime}=-A$
$\Rightarrow A=-A$
$\Rightarrow A+A=O$
$\Rightarrow 2 A=O$
$\Rightarrow A=O$
Therefore, A is a zero matrix.